Making a bone match another bone's rotation on a single local axis
Did you mean to link a file?

The simplest thing to do is try a copy rotation constraint in local space.  For what you're trying to do, this sounds like good enough.  Here's how to set it up:



You can see that the neck, even though it has no transformation, is following the Y axis rotation of the head, regardless of rotation of the torso.

If you rotate head, the neck will rotate with it; if you rotate torso, the neck will rotate with it as well.  That means that neck might rotate too much for your taste, but then, you really have to put more thought into how you want neck to behave.  You could always use transformation constraints or drivers to rescale any rotation to a certain desirable range.

I said this was the simplest thing to do.  Copying rotation in a single axis is dangerous, because Euler angles aren't really cleanly divisible into axes: by rotating a bone in its Z and X axes, you can create Y axis rotation, and there are places where one axis of rotation seems to turn into a different axis.  This can create unexpected behavior and poor interpolation.  Copying rotation from a single axis is really only safe when both bones involve only ever rotate on a single axis.

What's better is to think instead about where you want your neck to face, and to use a locked track constraint (locked in Y in this case) to face that point.  But that requires rethinking the problem slightly.